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Mirrors > Home > ILE Home > Th. List > znnen | Unicode version |
Description: The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.) |
Ref | Expression |
---|---|
znnen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unrab 3347 | . . 3 | |
2 | nnssz 9071 | . . . . . 6 | |
3 | dfss1 3280 | . . . . . 6 | |
4 | 2, 3 | mpbi 144 | . . . . 5 |
5 | dfin5 3078 | . . . . 5 | |
6 | 4, 5 | eqtr3i 2162 | . . . 4 |
7 | 6 | uneq1i 3226 | . . 3 |
8 | rabid2 2607 | . . . 4 | |
9 | elznn 9070 | . . . . 5 | |
10 | 9 | simprbi 273 | . . . 4 |
11 | 8, 10 | mprgbir 2490 | . . 3 |
12 | 1, 7, 11 | 3eqtr4ri 2171 | . 2 |
13 | nnex 8726 | . . . 4 | |
14 | 13 | enref 6659 | . . 3 |
15 | zex 9063 | . . . . . 6 | |
16 | 15 | rabex 4072 | . . . . 5 |
17 | nn0ex 8983 | . . . . 5 | |
18 | negeq 7955 | . . . . . . . 8 | |
19 | 18 | eleq1d 2208 | . . . . . . 7 |
20 | 19 | elrab 2840 | . . . . . 6 |
21 | 20 | simprbi 273 | . . . . 5 |
22 | negeq 7955 | . . . . . . 7 | |
23 | 22 | eleq1d 2208 | . . . . . 6 |
24 | nn0negz 9088 | . . . . . 6 | |
25 | nn0cn 8987 | . . . . . . . . 9 | |
26 | 25 | negnegd 8064 | . . . . . . . 8 |
27 | 26 | eleq1d 2208 | . . . . . . 7 |
28 | 27 | ibir 176 | . . . . . 6 |
29 | 23, 24, 28 | elrabd 2842 | . . . . 5 |
30 | elrabi 2837 | . . . . . . . 8 | |
31 | 30 | adantr 274 | . . . . . . 7 |
32 | 31 | zcnd 9174 | . . . . . 6 |
33 | 25 | adantl 275 | . . . . . 6 |
34 | negcon2 8015 | . . . . . 6 | |
35 | 32, 33, 34 | syl2anc 408 | . . . . 5 |
36 | 16, 17, 21, 29, 35 | en3i 6665 | . . . 4 |
37 | nn0ennn 10206 | . . . 4 | |
38 | 36, 37 | entri 6680 | . . 3 |
39 | inrab2 3349 | . . . 4 | |
40 | incom 3268 | . . . 4 | |
41 | rabeq0 3392 | . . . . 5 | |
42 | 0red 7767 | . . . . . . . 8 | |
43 | simpl 108 | . . . . . . . . 9 | |
44 | 43 | nnred 8733 | . . . . . . . 8 |
45 | nngt0 8745 | . . . . . . . . 9 | |
46 | 45 | adantr 274 | . . . . . . . 8 |
47 | nn0ge0 9002 | . . . . . . . . . 10 | |
48 | 47 | adantl 275 | . . . . . . . . 9 |
49 | 44 | le0neg1d 8279 | . . . . . . . . 9 |
50 | 48, 49 | mpbird 166 | . . . . . . . 8 |
51 | 42, 44, 42, 46, 50 | ltletrd 8185 | . . . . . . 7 |
52 | 42 | ltnrd 7875 | . . . . . . 7 |
53 | 51, 52 | pm2.65da 650 | . . . . . 6 |
54 | 53, 4 | eleq2s 2234 | . . . . 5 |
55 | 41, 54 | mprgbir 2490 | . . . 4 |
56 | 39, 40, 55 | 3eqtr3i 2168 | . . 3 |
57 | unennn 11910 | . . 3 | |
58 | 14, 38, 56, 57 | mp3an 1315 | . 2 |
59 | 12, 58 | eqbrtri 3949 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 crab 2420 cun 3069 cin 3070 wss 3071 c0 3363 class class class wbr 3929 cen 6632 cc 7618 cr 7619 cc0 7620 clt 7800 cle 7801 cneg 7934 cn 8720 cn0 8977 cz 9054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-er 6429 df-en 6635 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 df-mod 10096 df-dvds 11494 |
This theorem is referenced by: qnnen 11944 |
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